HERMITIAN POSITIVE SEMI-DEFINITE SOLUTION OF THE QUATERNION MATRIX EQUATION $ X^2+XB^*X+BX+Q=0 $

This paper discusses necessary and sufficient conditions for the existence of Hermitian positive semi-definite solutions of the quadratic matrix equation $ X^2+XB^*X+BX+Q=0\, (Q\geq0) $ on the quaternion field and its iterative solution method. Some necessary and sufficient conditions for the existence of Hermitian positive semi-definite solutions of this equation are proved mainly for the characteristics of coefficient matrices by introducing appropriate parameters to establish matrix inequalities and by using the theory of fixed points on convex sets. On this basis, three convergent iteration formats are constructed for different conditions and solution existence intervals, and the selection of the initial matrix is given according to each iteration property. The algorithm is solved in the MATLAB environment by using the complex operator of the quaternion matrix. At the same time, a perturbation analysis is carried out on the solution of the equation, and two perturbation error bounds are obtained by considering the Frobenius norm. Three numerical examples are used to test the effectiveness and feasibility of the given method.